Area of Julia sets of non-renormalizable cubic polynomials

TL;DR

This paper proves the existence of a non-renormalizable cubic polynomial with a Julia set of positive area, expanding the class of known rational maps with such properties beyond quadratic cases.

Contribution

It introduces a new approach to demonstrate that non-renormalizable cubic polynomials can have Julia sets with positive area, a previously unknown phenomenon.

Findings

Existence of non-renormalizable cubic polynomial with positive area Julia set

New method developed for analyzing Julia sets of cubic polynomials

Extends known examples beyond quadratic maps

Abstract

The long-standing problem of existence of nowhere dense rational Julia set with positive area has been solved by an example in quadratic polynomials by Buff and Ch\'eritat. Since then many efforts have been devoted to finding out new classes of rational maps with nowhere dense Julia sets having positive area. So far, all known examples of this kind are renormalizable with only one exception which is a quadratic polynomial. In this paper, by developing a new approach, we prove that there exists a non-renormalizable cubic polynomial having a Julia set with positive area.